There aren't a lot of different uses for 2D artists in the game industry these days, since only little indie teams make 2D games any more. In my experience ther are: Concept artists, UI artists, and occasionally Marketing artists. For concept art, there's much more demand for environment concept artists than character concept artists, sometimes they are a separate job.
If you want to do UI art, then have some nice UI art to show. I haven't done it myself, but I get the impression UI art tends to bleed into UI/UX design, so you may want to look into that stuff too, maybe have some interactive web mockups.
I've only seen a few, high-end jobs for marketing artists, and they're the same as illustrators as far as I know, so do some cover art style illustrations for your portfolio. Take a look at League of Legends splash art, etc.
Concept art is pretty broad. You'll want a couple character designs, definitely with some turnarounds (front, back, side views), some different environment concepts, some props (weapons, and close up environmental details, etc.), and possibly a vehicle or two. You could focus on just environments if you like, otherwise don't just focus on characters—everyone want's to do them, but they're only a small percentage of a game's content.
Most concept art these days seems to be either very "realistic", photobash/3D render stuff, or painted, sort of cartoony, Warcraft/Starcraft style stuff (or both) . . . and possibly a flatter, anime-influenced variant of the second category has gotten more populare lately. Don't let any personal stylistic ideas stray too far from these. Game studios don't pay you for personal work, and they won't want weird artsy ideas getting in the way of their art direction. Unless you are the best of the best of course.
The big thing is, people looking at your portfolio will assume (correctly) that you are incapable of doing anything that you haven't shown.